论文标题
Gabriel-Krull维度和最小原子类别
Gabriel-Krull dimension and minimal atoms in Grothendieck categories
论文作者
论文摘要
令$ \ Mathcal {a} $为Grothendieck类别。在本文中，我们通过$ \ Mathcal {A} \ Mathrm {spec} \，\ Mathcal {a a} $的开放子集对半noetherian类别的本地化子类别进行分类。对于半noetherian本地相干类别$ \ Mathcal {a} $，我们在$ \ Mathcal {a} \ Mathrm {spec} \，\ Mathcal {a a} $上引入了一个新的拓扑，我们证明了它对Ziegler spectrum $ \ mathrm {Z____________________________________此外，对于本地连贯的类别$ \ Mathcal {a} $，我们提供了一个新的特征，以定位有限类型的$ \ Mathcal {a} $的子类别。我们使用预订$ \ leq $在$ \ Mathcal {a} \ Mathrm {spec} \，\ Mathcal {a} $上定义对象的维度，该对象是Gabriel-Krull Dimension的下限。最后，我们研究了$ \ noterian对象$ m $ in $ \ mathcal {a} $的最小原子，并为$ m $ $ m $的最小原子数的有限数建立了足够的条件。
Let $\mathcal{A}$ be a Grothendieck category. In this paper, we classify localizing subcategories of a semi-noetherian category $\mathcal{A}$ through open subsets of $\mathcal{A}\mathrm{Spec}\,\mathcal{A}$. For a semi-noetherian locally coherent category $\mathcal{A}$, we introduce a new topology on $\mathcal{A}\mathrm{Spec}\,\mathcal{A}$ and we show that it is homeomorphic to the Ziegler spectrum $\mathrm{Z_g}\mathcal{A}$. Moreover, for a locally coherent category $\mathcal{A}$, we provide a new characterization of localizing subcategories of finite type of $\mathcal{A}$. We define a dimension for objects using the preorder $\leq$ on $\mathcal{A}\mathrm{Spec}\,\mathcal{A}$, which serves as a lower bound of the Gabriel-Krull dimension. Finally, we investigate the minimal atoms of a noetherian object $M$ in $\mathcal{A}$ and establish sufficient conditions for the finiteness of the number of minimal atoms of $M$.
